As a kid I was tempted to shoot gulls, and was always told by my father that they were protected because they are such an efficient scavenger. True or not, they don't cause a lot of problems inland, are certainly part of the natural ecology, and certainly are not edible, so why bother them? I can see how Maritimers with piles of gull crap on their roofs might feel differently.
Songbirds are protected for a good reason...most species are suffering serious declines in population, from habitat loss and other reasons...which brings us to the perfect birds to shoot: the introduced species that are one of the major reasons for the decline in native species.
House sparrows, starlings and pigeons (rock doves) are all introduced pest species that displace and compete with natives. They are usually legal to shoot anytime, year-round, no limit. Check local regs to make sure.
Crows are usually legal to shoot at any time. They seem to be smarter than most people, so it's doubtful they are in much danger of depletion.
And finally...just to stir the pot a bit...don't forget the ultimate pests, the big-game of introduced, invasive, ecologically-disruptive, shoot-on-sight, PITA critters. No, it's not wild pigs, at least not in Canada yet. It's feral cats.